51.

The curve y = (λ + 1)x² + 2 intersects the curve y = λx + 3 in exactly one point, if λ equals -

• {–2, 2}

• {1}

• {-2}

• {-2}

If the curves y² = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is

• 9/2

• 6

• 7/2

• 4

If L₁ is the line of intersection of the plane 2x – 2y + 3z – 2 = 0, x – y + z + 1 = 0 and L₂ is the line of intersection of the plane x + 2y – z – 3 = 0, 3x – y + 2z – 1 = 0, then the distance of the origin from the plane containing the lines L₁
and L₂ is :

• $\frac{1}{\sqrt{2}}$

• $\frac{1}{4\sqrt{2}}$

• $\frac{1}{3\sqrt{2}}$

• $\frac{1}{2\sqrt{2}}$

The equation of the plane through (1, 2,- 3) and (2,- 2, 1) and parallel to X-axis is

• y - z + 1 = 0

• y - z - 1 = 0

• y + z - 1 = 0

• y + z + 1 = 0

Three lines are drawn from the origin O with direction cosines proportional to (L,-1, 1), (2,-3, 0) and (1, 0, 3). The three lines are

• not coplanar

• coplanar

• perpendicular to each other

• coincident

A straight line joining the points (1, 1, 1) and (0, 0, 0) intersects the plane 2x + 2y + z = 10 at

• (1, 2, 5)

• (2, 2, 2)

• (2, 1, 5)

• (1, 1, 6)

Angle between the planes x + y + 2z = 6 and 2x - y + z = 9 is

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The value of $\mathrm{\lambda }$ for which the straight line $\frac{\mathrm{x} - \mathrm{\lambda }}{3} = \frac{\mathrm{y} - 1}{2 + \mathrm{\lambda }} = \frac{\mathrm{z} - 3}{- 1}$ may lie on the plane x - 2y = 0, is

• 2

• 0

• $- \frac{1}{2}$

• $\mathrm{there} \mathrm{is} \mathrm{no} \mathrm{such} \mathrm{\lambda }$

In a triangle PQR, $\angle \mathrm{PQR}, \angle \mathrm{R} = . \mathrm{If} \tan \left(\frac{\mathrm{P}}{2}\right) \mathrm{and} \tan \left(\frac{\mathrm{Q}}{2}\right)$ are roots of ax² + bx + c = 0, where a$\ne$ 0, then which one is true?

• c = a + b

• a = b + c

• b = a + c

• b = c

If C is the reflection of A (2, 4) in x-axis and B is the reflection of C in y-axis, then | AB | is

• 20

• 2√5

• 4√5

• 4